Problem: Simplify and expand the following expression: $ \dfrac{a + 5}{a + 5}-\dfrac{3a - 10}{4a - 6} $
Solution: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(a + 5)(4a - 6)$ Multiply the first term by $\dfrac{4a - 6}{4a - 6}$ $ \begin{align*} \dfrac{a + 5}{a + 5} \times \dfrac{4a - 6}{4a - 6} & = \dfrac{(a + 5)(4a - 6)}{(a + 5)(4a - 6)} \\ & = \dfrac{4a^2 + 14a - 30}{(a + 5)(4a - 6)}\end{align*} $ Multiply the second term by $\dfrac{a + 5}{a + 5}$ $ \begin{align*} \dfrac{3a - 10}{4a - 6} \times \dfrac{a + 5}{a + 5} & = \dfrac{(3a - 10)(a + 5)}{(4a - 6)(a + 5)} \\ & = \dfrac{3a^2 + 5a - 50}{(4a - 6)(a + 5)}\end{align*} $ Now we have: $ = \dfrac{4a^2 + 14a - 30}{(a + 5)(4a - 6)} - \dfrac{3a^2 + 5a - 50}{(4a - 6)(a + 5)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{4a^2 + 14a - 30 - (3a^2 + 5a - 50)}{(a + 5)(4a - 6)} $ $ = \dfrac{4a^2 + 14a - 30 - 3a^2 - 5a + 50}{(a + 5)(4a - 6)} $ $ = \dfrac{a^2 + 9a + 20}{(a + 5)(4a - 6)}$ Expand the denominator: $ = \dfrac{a^2 + 9a + 20}{4a^2 + 14a - 30}$